Radio numbers for generalized prism graphs

A radio labeling is an assignment $c:V(G) \rightarrow \textbf{N}$ such that every distinct pair of vertices $u,v$ satisfies the inequality d(u,v)+|c(u)-c(v)|\geq \diam(G)+1. The span of a radio labeling is the maximum value. The radio number of $G$, $rn(G)$, is the minimum span over all radio labelings of $G$. Generalized prism graphs, denoted $Z_{n,s}$, $s \geq 1$, $n\geq s$, have vertex set $\{(i,j)\,|\, i=1,2 \text{and} j=1,...,n\}$ and edge set $\{((i,j),(i,j \pm 1))\} \cup \{((1,i),(2,i+\sigma))\,|\,\sigma=-\left\lfloor\frac{s-1}{2}\right\rfloor\,\ldots,0,\ldots,\left\lfloor\frac{s}{2}\right\rfloor\}$. In this paper we determine the radio number of $Z_{n,s}$ for $s=1,2$ and $3$. In the process we develop techniques that are likely to be of use in determining radio numbers of other families of graphs.Comment: To appear in Discussiones Mathematicae Graph Theory. 16 pages, 1 figur

Improved Bounds for Radio k

A number of graph coloring problems have their roots in a communication problem known as the channel assignment problem. The channel assignment problem is the problem of assigning channels (nonnegative integers) to the stations in an optimal way such that interference is avoided as reported by Hale (2005). Radio k-coloring of a graph is a special type of channel assignment problem. Kchikech et al. (2005) have given a lower and an upper bound for radio k-chromatic number of hypercube Qn, and an improvement of their lower bound was obtained by Kola and Panigrahi (2010). In this paper, we further improve Kola et al.'s lower bound as well as Kchikeck et al.'s upper bound. Also, our bounds agree for nearly antipodal number of Qn when n≡2 (mod 4)

The Radio Number of Grid Graphs

The radio number problem uses a graph-theoretical model to simulate optimal frequency assignments on wireless networks. A radio labeling of a connected graph $G$ is a function $f:V(G) \to \mathbb Z_{0}^+$ such that for every pair of vertices $u,v \in V(G)$, we have $\lvert f(u)-f(v)\rvert \ge \text{diam}(G) + 1 - d(u,v)$ where $\text{diam}(G)$ denotes the diameter of $G$ and $d(u,v)$ the distance between vertices $u$ and $v$. Let $\text{span}(f)$ be the difference between the greatest label and least label assigned to $V(G)$. Then, the \textit{radio number} of a graph $\text{rn}(G)$ is defined as the minimum value of $\text{span}(f)$ over all radio labelings of $G$. So far, there have been few results on the radio number of the grid graph: In 2009 Calles and Gomez gave an upper and lower bound for square grids, and in 2008 Flores and Lewis were unable to completely determine the radio number of the ladder graph (a 2 by $n$ grid). In this paper, we completely determine the radio number of the grid graph $G_{a,b}$ for $a,b>2$, characterizing three subcases of the problem and providing a closed-form solution to each. These results have implications in the optimization of radio frequency assignment in wireless networks such as cell towers and environmental sensors.Comment: 17 pages, 7 figure

Radio Graceful Labelling of Graphs

Radio labelling problem of graphs have their roots in communication problem known as \emph{Channel Assignment Problem}. For a simple connected graph $G=(V(G), E(G))$, a radio labeling is a mapping $f \colon V(G)\rightarrow \{0,1,2,\ldots\}$ such that $|f(u)-f(v)|\geq {\rm diam}(G)+1-d(u,v)$ for each pair of distinct vertices $u,v\in V(G)$, where $\rm{diam}(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$. A radio labeling $f$ of a graph $G$ is a \emph{radio graceful labeling} of $G$ if $f(V(G)) = \{0,1,\ldots, |V(G)|-1\}$. A graph for which a radio graceful labeling exists is called \emph{radio graceful}. In this article, we study radio graceful labeling for general graphs in terms of some new parameters

Optimal radio labellings of complete m-ary trees

A radio labelling of a connected graph G is a mapping f : V (G) &rarr; {0, 1, 2, ...} such that | f (u) - f (v) | &ge; diam (G) - d (u, v) + 1 for each pair of distinct vertices u, v &isin; V (G), where diam (G) is the diameter of G and d (u, v) the distance between u and v. The span of f is defined as maxu, v &isin; V (G) | f (u) - f (v) |, and the radio number of G is the minimum span of a radio labelling of G. A complete m-ary tree (m &ge; 2) is a rooted tree such that each vertex of degree greater than one has exactly m children and all degree-one vertices are of equal distance (height) to the root. In this paper we determine the radio number of the complete m-ary tree for any m &ge; 2 with any height and construct explicitly an optimal radio labelling.<br /